A class of stochastic partial differential equations for interacting superprocesses on a bounded domain

Abstract

A class of interacting superprocesses on $\mathbb{R}$, called superprocesses with dependent spatial motion (SDSMs), were introduced and studied in Wang [32] and Dawson et al. [9]. In the present paper, we extend this model to allow particles moving in a bounded domain in $\mathbb{R}^{d}$ with killing boundary. We show that under a proper re-scaling, a class of discrete SPDEs for the empirical measure-valued processes generated by branching particle systems subject to the same white noise converge in $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$ to the SPDE for an SDSM on a bounded domain and the corresponding martingale problem for the SDSMs on a bounded domain is well-posed.